Tools for young mathmaticians

Manipulatives

Why Use Manipulatives?

Manipulatives allow students to interpret, comprehend and represent a wide variety of math concepts.

Levels of Abstraction

Each manipulative has different levels of difficulty and understanding, which we refer to as a “level of abstraction”. This phrase refers to the level of abstract thinking that is required by a student to successfully use the particular manipulative. For instance, can they understand and explain why a particular block holds a representational value (one blue block has a value of 9) even though it is only one block with no marks on it?

When adding manipulatives to your math curricula, allow students to explore the manipulatives before being asked to solve problems with them. We find daily transition times are great opportunities for students to explore.

The list below provides a progression for the levels of abstraction with the most commonly used manipulatives. These manipulatives are excellent for even the youngest mathematicians! Just keep in mind your assessments should note the difficulty or “level of abstraction” that each manipulatives requires of a student.

Note: We do not sell manipulatives, nor do we have any relationships with manufacturers or vendors. This list is meant solely as a resource.

Common Manipulatives

Wooden Cubes

Typically made from wood, these cubes are ideal for counting activities, patterns, beginning addition, subtraction, multiplication and division, pentomino building as well as explorations with volume and surface area. Their one-inch size makes them ideal for small children to use. These one-to-one blocks have a low level of abstraction.

Color Tiles

Usually made from colored plastic, these one-inch squares are ideal for working on area and perimeter tasks as well as counting, sorting, patterns and pentominos. These one-to-one blocks have a low level of abstraction

Unifix Cubes

Unifix cubes are the basic block for any classroom. They are made of plastic and connect to each other on two opposing sides. They can be used to teach almost all math concept areas, ranging from one-to-one correspondence, patterns and basic number operations to fractions, multi-base projects and beginning algebra. Since students can separate unifix cubes into single units, the level of abstraction of these manipulatives is very low. When students are working to solve a tricky concept and having difficulty showing their thinking with a different (more difficult) manipulative, take them back to these unifix cubes and let them create their model with these one-to-one blocks.

Centimeter Cubes

These small plastic cubes each measure one cubic centimeter and do not connect with each other. They can be used for counting, geometry and also in conjunction with Cuisenaire rods and base ten blocks as those manipulatives are also based on a centimeter unit. These blocks are small so consider using them to help develop fine motor skills with some students as they work to solve math tasks. Inversely, if a student is thinking faster than his/her motor skills will move, provide him/her with larger wooden cubes or unifix blocks. Because these blocks can be broken into single units, they have a low level of abstraction.

Multilinks

Multilinks are similar to unifix cubes with the exception that students are able to connect these cubes to each other on all sides of each block. Multilinks also have triangular prism pieces that connect to the other blocks. While Multilinks also have a low level of abstraction, they lend themselves to geometry more easily than unifix cubes as students can make multi-dimensional shapes. If you have a student who is struggling to demonstrate their thinking through the use of another manipulative, give them these blocks to more easily illustrate their work.

Centimeter Cubes with Connectors

These small plastic cubes measure one cubic centimeter and are able to connect to one another. They are useful in the areas of counting, place value and geometry activities. These blocks require a higher level of fine motor dexterity than the non-connecting centimeter cubes, as students have to apply strong pressure to connect the units together. If a student's thinking is outpacing his/her motor skills, provide him/her with larger wooden blocks or unifix cubes. Because these blocks can be broken into single units, they have a low level of abstraction.

Base Ten Blocks

Base ten blocks are, at times, referred to as Dienes blocks. Originally they were made from wood, and now plastic sets come with two colors, usually red and blue. Dual-color sets are well suited for projects with integers, where students may use one color for negative integers and the other for positive. These blocks are another must have for classrooms. Each set is based on a centimeter unit and contains units, rods, flats and cubes. When students connect ten of the units, their solution is the same size as a rod (10 units). When students put ten rods together, they can trade for a flat (equal to 100 units), and stacking ten flats together creates a cube (equal to 1000 units). With young children, use these blocks for building structures, counting, and beginning trading. Learning base 10 place value, addition, subtraction, multiplication, and division are also spectacular uses for this manipulative. Use them for building arrays for multiplication and division as well as operations with integers and beginning algebra. Base ten blocks have a medium level of abstraction. While students must assume a value for a block, (one rod is 10, even though it is just one block), unlike the Cuisenaire rods, base ten blocks provide centimeter marks on the each block so students can easily double check their values for each block.

Pattern Blocks

Pattern blocks are a basic necessity for every classroom. Among the shapes within each set are a hexagon, trapezoid, triangle, parallelogram, square and rhombus. Most pieces have relationships to one another. For example, assuming the triangle has a value of 1, the parallelogram then holds a value of 2, the trapezoid a value of 3 and the hexagon a value of 6. The square and the rhombus values do not translate as easily but make for excellent investigations. Pattern blocks are a very versatile manipulative. Use for symmetry, counting, money values, geometry, angles, fractions (what if the hexagon was the whole?) and multi-base projects. These blocks can facilitate a very high level of abstraction, as students are required to associate a value with a block shape. (One yellow hexagon can have a value of 6 even though it's just one block). If students are not yet demonstrating the ability to make that leap, you need to have them work with a less complicated manipulative to solve their task.

Pentablocks

These lightweight Pentablocks with opaque washed surfaces are a wonderful extension to pattern block activities. They do NOT share the same measurements as the pattern blocks, yet they offer brilliant extensions to tasks learned with the pattern blocks, using an entirely new set of acute and obtuse angles to measure, as well as new symmetry design challenges.

Cuisenaire Rods

These wooden Cuisenaire rods are graduated in length and based on a single centimeter cube as a unit of measure. They are one of the oldest manipulatives around, developed decades ago, and also one of the most versatile tools in a classroom. Cuisenaire rods can be used for counting (just counting rods without attaching a value), addition, subtraction, multiplication (multiples and factors), division, area, perimeter, volume, geometry and algebra. These rods are complex in nature and thus have a very high level of abstraction, meaning that, for ideal use, students must be able to assign values to different sized single blocks. (One blue block has a value of nine units even though it's just one block). Remember, if students are having difficulty illustrating their work with these Cuisenaire blocks, have them use unifix cubes instead. The level of abstraction for that manipulative is lower, and students will be able to prove and illustrate their work more easily using them.

Geoblocks

These wooden Geoblocks come in a set filled with a myriad of geometric shapes. They are certainly used with young children for building and vocabulary development (always use rich mathematical vocabulary with your students), but more frequently they are used with geometry projects. Students decide upon a square unit of measure and then work to find the volume and surface area for these regular and irregular shapes. They construct surface area nets (or jackets) for the individual shapes as well as combine several blocks to make the learning task more complex.

Fraction Tiles

Every classroom should have a few sets of fraction tiles. These make for possibly the best way to finally understand adding, subtracting, multiplying and dividing fractions! Each set comes with 16ths, 12ths, 8ths, 6ths, 4ths, 3rds, halfs, and a whole. Of course, it's more challenging and fun to create projects that change the value of the "whole," thus changing the value of the other pieces! Young children use these transparent tiles to begin seeing pieces that are larger and smaller than the others, greater/less than comparisons, then move to equivalency and into basic number operations. This is an outstanding manipulative to use in your classroom.

Fractional Pattern Blocks

These small tiles are sized to work with pattern blocks. They are an outstanding addition to any pattern block set as they let you turn a hexagon into fourths and twelfths. These make a great extension piece for fraction work.

Fractional Pattern Blocks

These pink and black wooden blocks measure exactly the same as pattern blocks. They are designed to offer students extensions with pattern block fractions. Basically, by assigning values to these pieces (what if one black piece had a value of one?), students can work through more complex fraction tasks. While they are convenient, students can do the same extensions with the standard pattern blocks.

Cuisenaire Rod Counting Track

These plastic counting tracks come in 24 inch and 36 inch lengths. They fit Cuisenaire rods and any other centimeter size cubes and have numbers and centimeter marks along the side. By placing blocks in the track, students can figure out the sum of a multi-digit equation or use the track to double check their answer for an algorithm they are learning. These are wonderful tools for students who can build complex equations but do not yet have the algorithm skills to solve the corresponding numeric equation.

Base Six Blocks

These plastic blocks mirror the base ten set with cubic centimeter units and rods, flats and cubes, except only six units form a rod. The flat, therefore, is equal to six rods (36 units), and the cube can be formed using six flats (216 units). Of course using multi-base blocks is a terrific extension for students needing to solidify their understanding of place value trading. Surprisingly, this manipulative is also great for younger students who are just learning to count and trade. Base ten might be too many objects to work with so teach the trading process using smaller bases. Other base block sets available are base 5, base 4, base 3 and base 2. Be aware, the smaller the base the more frequent the trades...extra practice, but more tricky too. Try base 2 yourself for a fun challenge!

Base Five Blocks

These plastic blocks mirror the base ten set with cubic centimeter units and rods, flats and cubes, except only five units form a rod. The flat, therefore, is only five rods in width (25 units) and the cube is five flats tall (125 units). Of course using multi-base blocks is a terrific extension for students needing to solidify their understanding of place value. Surprisingly, this manipulative is also great for younger students who are just learning to count and trade. Base ten counting might involve too many objects to work with so teach the trading process using smaller bases. Other base block sets available are base 6, base 4, base 3 and base 2. Be aware, the smaller the base the more frequent the trades...extra practice, but more tricky too. Try base 2 yourself for a fun challenge!

Base Four Blocks

These plastic blocks mirror the base ten set with cubic centimeter units and rods, flats and cubes, except only four units form a rod. The flat, therefore, is only four rods in width (16 units), and the cube is four flats tall (64 units). Of course using multi-base blocks is a terrific extension for students needing to solidify their understanding of place value. Surprisingly, this manipulative also great for younger students who are just learning to count and trade. Base ten blocks might involve too many pieces to work with so teach the trading process using smaller bases. Other base block sets available are base 6, base 5, base 3 and base 2. Be aware, the smaller the base the more frequent the trades. Extra practice, but lots more tricky too. Try base 2 yourself for a fun challenge!

Base Three Blocks

These plastic blocks mirror the base ten set with cubic centimeter units and rods, flats and cubes, except only three units form a rod. The flat, therefore, is only three rods in width (9 units) and the cube is three flats tall (27 units). Of course using multi-base blocks is a terrific extension for students needing to solidify their understanding of place value. Surprisingly, this manipulative is also great for younger students who are just learning to count and trade. Base ten blocks might involve too many numbers to work with so teach the trading process using smaller bases. Other base block sets available are base 6, base 5, base 4 and base 2. Be aware, the smaller the base the more frequent the trades...extra practice, but more tricky too. Try base 2 yourself for a fun challenge!

Base Two Blocks

These plastic blocks mirror the base ten set with cubic centimeter units and rods, flats and cubes except only two units form a rod. The base two flat, therefore, is only two rods in width (4 units), and the cube is two flats tall (8 units). Of course using multi-base blocks is a terrific extension for students needing to solidify their understanding of place value. Surprisingly, this manipulative is also great for younger students who are just learning to count and trade (look at base 5 or 6, as base 2 is the most difficult). Base ten might be too many numbers to work with so teach the trading process using smaller bases. Other base block sets available are base 6, base 5, base 4 and base 3. Be aware, the smaller the base the more frequent the trades...extra practice, but lots more tricky too. Try base 2 yourself for a fun challenge!

Tangrams

These plastic tangram sets can be purchased with large, medium or small pieces. Each set has seven pieces. The basic tangram puzzle requires the user to use all seven pieces to construct a square. There are a wide variety of puzzle activities that are available to use with the tangram pieces. It's also fun to have kids use one piece of paper and fold their own tangram pieces (patterns on the internet). We use tangrams for projects with fractions, area, perimeter and angles.

Geoboards

Geoboards come in both plastic and wood models and vary in size. Most geoboards are a series of pegs laid out in a vertical and horizontal pattern. You can also purchase geoboards with a circular peg pattern. Young children use them to explore making geometric shapes with rubber bands. They are often used for learning to figure out area and perimeter (the smallest square you can make would have a perimeter of 4 and an area of 1). Older students begin learning to build polygons and then find ways to determine the area of these irregular shapes. In this picture, the green band is the original polygon and the yellow band is placed in a way to start figuring out the area of the overall shape. Geoboards can also be used for learning fractions and graphing/grid work. If you place four geoboards together, you can create four quadrants (with X and Y axes and positive and negative areas) and create a Cartesian plane for work with grids.

Dice

Have a wide variety of dice on hand to add to all the manipulatives you are using. Repetition of newly developing skills is important for learning. Use dice to turn that practice into games and make that repetition more fun for your students. Notice that there are dice with numerals though 32 as well as dice with operations symbols, fractions and roman numerals. Having a variety of dice allows you as the teacher to individualize the stretch you offer each student as they practice.

Stamps & Stencils

It's important to have a variety of tools available for students to use when documenting their work. Many children (especially young ones) can think in more detail and complexity than they can write. Having stamps available for illustrating their work helps the learning process. There are stamps for pattern blocks, base blocks, Cuisenaire blocks, grids, and geoboards to name a few. In addition, there are also templates available for tracing the shapes for many of those manipulatives as well. We show a Cuisenaire template in this photo. For some students, tracing shapes is good practice for developing small motor skills. Some schools have access to a die cutting machines. If you are one of those schools, you are in luck because there are die cuts available for most of the manipulative block sets. Having paper cutouts available for students to illustrate their thinking process is cost saving as well as fun.

Pentominoes

These plastic blocks come in a variety of shapes, all based on the different positions of five squares laid next to one another. Take five square pieces of paper or five cubes and see how many DIFFERENT shapes you can make when each full side of a square touches another full side of a square. You will quickly get into having students make shape rotations and flips to see if their shape is the same as another one they have already discovered.

Dime Solids

Dime solids are three dimensional foam blocks. They are wonderful for beginning work with volume and surface area.

Algeblocks

These plastic blocks are similar in size to the base blocks (centimeter unit as the "base"), but they provide both 'x' and 'y' axes. Shown in the picture, the yellow blocks could be 1 times x, x squared, and x cubed. The dark orange could be 1 times y, y squared and y cubed. The magic is that the light orange is then x times y and xy cubed. Algeblocks facilitate learning to add, multiply, divide and work with integers while using variables.